On weak \(P\)-coherent rings (Q6650381)

Let \(R\) be a commutative ring. In the paper under review, the authors introduced a generalization of the well-known notion of weak coherent rings, which they called weak P-coherent rings. A ring \(R\) is called weak P-coherent, whenever \(P\subseteq J\) are ideals of \(R\) where \(P\) is a finitely generated prime ideal and \(J\) is finitely presented, then \(P\) is finitely presented. They investigated the stability of this property under direct products and homomorphic image, and its transfer to various contexts of constructions such as trivial ring extensions and amalgamated algebras along an ideal. Among others, they proved that a finite product of rings \(R_{i}\) is a weak P-coherent ring if and only if each \(R_{i}\) is a weak P-coherent ring; and if \(R\) is a weak P-coherent ring and \(I\) is a finitely presented ideal of \(R\), then \(R/I\) is a weak P-coherent ring. For the amalgamated algebras along some ideals of \(R\), they proved that for a finitely presented ideal \(J\) of \(f(A)+J\), if \(A\bowtie^{f} J\) is weak P-coherent, then so \(A\) is; and if \(f^{-1}(J)\) is a finitely presented ideal of \(A\) and \(J\) is a finitely generated ideal of \(f(A)+J\), then \(A\bowtie^{f} J\) weak P-coherent implies that \(f(A)+J\) is P-coherent.